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Strain compatibility equation in non-linear solid mechanics!!!

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We have six strain compatibility equations, which are obtained from strain-displacement relations by making an assumptions 'small strains'. Strain compatibility equations ensure a single valued and continuous displacemnet filed. These equations are used in stress based approach.

Now my queries are as following.

[1] Do we have strain compatibility equations for non-linear strain-displacement relations?

[2] Do we follow stress based approach in non-linear solid mechanics.

For me it looks like it is difficult (may not be possible also) derive strain compatibility equations in nonlinear solid mechanics.

I request that somebody may through light on the above by giving main focus on 'Strain compatibility equations in nonlinear solid mechanics'

Comments

First note that the classical compatibility equations are necessary and sufficient only for simply-connected domains. If your body has "holes", i.e. for multiply-connected domains you need some extra conditions. So, let's worry about local compatibility equations as the global compatibility is much more complicated and strongly depends on the topology of the domains of interest.

In linearized elasticity, starting from strain tensor, you want to calculate a single displacement vector at a point B having the displacement at a fixed point A. You would do exactly the same thing in nonlinear elasticity. Instead of those six compatibility equations you have "Curl F = 0", where F is the deformation gradient. Note that here it is assumed that the ambient space is Euclidean.

You can equivalently write the compatibility equations in terms of the right Cauchy-Green strain tensor C. Thinking of C as the pull-back of the (Euclidean) metric of the ambient space, C is compatible if and only if its curvature tensor is zero. In dimension three, this gives six compatibility equations. However, keeping in mind that there are three Bianchi identities satisfied by the curvature tensor there are finally three independent compatibility equations (this is the rigorous version of the vague arguments you see in elasticity books).

The unfortunate tradition is to go from linearized elasticity to nonlinear elasticity. This misleading practice has caused much confusion in the literature. The natural approach is to start from nonlinear elasticity and then linearize about a given motion if necessary. Of course, there are infinitely many linearized theories, in general. So, seeing that there are displacement and stress-based formulations in linear elasticity doesn't mean that one should expect the same thing in nonlinear elasticity. In linear elasticity, in certain problems (e.g. if all the boundary conditions are written in terms of tractions) it may be convenient to write everything, and in particular, compatibility equations in terms of stresses. Having a linear relation between stress and strain tensors, this is a simple task. In the nonlinear theory, compatibility equations are written in terms of F or C. Depending on the stress-strain relations (coming form a strain energy or free energy density) you may be able to write everything in terms of stress, though this may not simplify the calculations at the end.

I could not completely understand in what sense you meant the equivalence between the (local) compatibility of the deformation 'gradient' and the right Cauchy-Green tensor fields.

Here are some of my thoughts related to the discussion that some might find useful:

I will also restrict the whole discussion to local existence or to simply-connected domains.

While it is true that if an invertible second order tensor field F is compatible (curl F = 0) then its Right Cauchy-Green field C = F^t F is compatible (Riemann-Christoffel (RC) curvature tensor vanishes), it is not true that if the C field is compatible, in the sense that RC vanishes, that the F field is compatible.

The easiest way to see this is to observe that if two deformations have identical C fields then the deformations can at most differ by a rigid deformation. Thus, a compatible C field defines a unique rotation field up to an inconsequential spatially uniform orthogonal tensor (R T Shield gives an elegant proof of this based on properties of harmonic functions, SIAM J. Appl. Math, 25(3), 1973, but there is also a more plug and chug pedestrian proof based on Christoffel symbols etc. that I shall post). Therefore, do a polar decomposition of the given F field, F = RU. Let its C = UU field be compatible (RC vanishes). This C field generates a 'unique' rotation field, say R*, such that R*U can be written as the deformation gradient of a position field. Now, if the R field of F is not identical to the R* field (up to a spatially uniform orthogonal tensor field) then F cannot be compatible. In fact, in the continuum theory of dislocations, this is how one can have non-trivial Nye tensor fields with zero elastic strain energy/stress.

Of course, even at small strains the proof of strain compatibility relies on constructing this 'unique' infinitesimal rotation field corresponding to a compatible small strain field.

A comment: as for compatibility proofs, the deformation/displacement 'gradient' arguments are the simplest, followed by small strain compatibility, followed by C compatibility. The mother of all compatibility arguments is that of B (left Cauchy Green compatibility) where, to my knowledge, a necessary and sufficient condition in 3-d does not exist today. Janet Blume did the 2-d case, and I did a suffcient condition in 3-d.

I shall post some class notes for the interested student (they were primarily to aid me in lecturing, but I think they may be useful for students willing to figure a few things out) and the paper on B-compatibility. Unfortunately, I will have to make a separate post for that as one cannot attach to comments.

Thanks for the interesting comments and also for your interesting J. Elasticity paper.

I agree with your first comment but am not completely clear on what you're trying to say. There is a theorem that says if curvature tensor of the manifold (M,C_{AB}) vanishes (M is the reference manifold and C_{AB} is the "flattened" right Cauchy-Green tensor as C as a map from tangent space of M to itself has components C^A{}_B) then locally there is a deformation mapping that has C as its right Cauchy-Green tensor. Having C, there is more than one F corresponding to C and this makes even discussing compatibility of F ambiguous. Is this what you meant? If yes, then what do you mean by " the F field " in your fourth paragraph?

Regarding the B tensor, I don't understand why there should be anything special about it compared to C. I had a quick look at your paper and got confused with where B is defined. My understanding is that given a point x in the ambient space B is a linear map from tangent space of the ambient space at this point to itself. On page 97, you define B on the reference configuration. Am I missing something?

C on the other hand is a map from tangent space of the material manifold at a point X to itself. It can be shown that C is the pull-back of the metric of the ambient space by the deformation map. Now if the ambient space is Euclidean its curvature tensor vanishes. Curvature tensor of the manifold (M,C) is the pull-back of this curvature tensor and should vanish too. Of course, the converse is harder to prove.

My guess is that one can do something similar by pushing forward the curvature of the material manifold (assuming that there are no defects, etc).

1) Suppose one is given an invertible second order tensor field A on a simply connected reference configuration, with points in the configuration generically marked by x.

Let the Riemann Christoffel curvature tensor components formed from the metric tensor components, the latter identified with the rectangular Cartesian components of the field A^t A, vanish.

Then, while there is a deformation y of the reference configuration such that (dy/dx)^t (dy/dx) = A^t A, it is not necessary that there exists *any* deformation z such that dz/dx = A.

In this sense, the question of Right Cauchy Green compatibility is not equivalent to the question of 'deformation gradient' compatibility.

2) I don't think you are missing anything with regard to the B compatibility question, but I do not understand :) (I think it is time we talked in person!) where the confusion is as the definition 1) on page 97 is the standard definition. Would it help if I had written B(y(x)) instead of B(x) and the transpose as as (dy/dx)^t (y(x))? Btw, I agree with you that geometrically B is defined on the tangent space of base points on the deformed manifold and so is the transpose. But here, since we are talking about an injective y and we are in Euclidean space so that tangent spaces on the deformed and undeformed manifolds at corresponding base points can be identified with each other, it really does not matter for the main existence question in hand.

So the main question of B compatibility is the natural one - given a positive definite, symmetric, second order tensor field B as a function of x, the question is to find conditions on this field such that a deformation y can be constructed such that (dy/dx)(dy/dx)^t at each x is equal to the B value at that x.

And, of course, most people who do continuum mechanics (including me) feel at first that B compatibility should not be any different than C compatibility - but the devil here very much seems to be in the details....

Also, Blume, Duda and Martins are all very capable people; I believe this question was also Blume's PhD work with Sternberg, so Sternberg would also have looked at this problem (at least to see whether there is a problem here at all or not).... But this should not deter you to try to answer the question, especially by reducing it to the C compatibility question somehow - I would be very interested in that.

2) Yes, if you write B(y(x)) things would be more consistent. Perhaps, in what you were doing this wouldn't make a difference but I think keeping the referential and spatial quantities completely clear may resolve some of the existing confusions.

Again, I feel that B and C compatibilities should be closely related. Let me read Blume's paper carefully and then get back to you.

Will look forward to hearing about any new additions you can make to answering the B compatibility question.

Some comments:

1) If one had the deformation y in hand, then keeping the reference and spatial objects separate can be done in 'constructive' fashion in the sense that one actually has spatial tangent spaces in hand etc. But when the spatial configuration itself is vaporware till you have actually constructed the deformation/have an existence guarantee for it, which is the main question here, maintaining that difference a priori seems to me to be a grey area. It is interesting that what you suspect as the main cause for confusion (actually there is no confusion in the question - there simply is no explicit answer in more than the plane case!), I find to be mere geometric window-dressing! Perhaps you will prove me wrong. That said, you will see that my formulation of the problem is actually the 'passive' geometric one, i.e. in charts (eqns. 3 thru 8 of my paper) :), which conceptually maintains two charts with the corresponding objects.

Anyway, keep in mind that the answer ultimately has to be in terms of some conditions on the field B on the only known configuration at hand - the reference. Also, that whatever compatibility condition you come up with, it has to be independent of the deformation y.

2) Blume punts on the existence question in 3-d. The theorem I provide would actually apply to her formulation for the Polar Decomposition rotation if you append the algebraic constraint Q^T Q = I to her eqns. 3.13.

Basically the answer comes down to wanting to apply the Froebenius theorem once you have formulated the problem as a system of total differential equations/Pfaffian forms with the complications that there are algebraic constraints and that there is no way of guaranteeing, just based on having the B field and the reference configuration, that the integrability condition can be made to hold. So you look for the next best thing to do.......

I assume by "nonlinear solid mechanics" you mean "nonlinear elasticity". In my opinion, the book by M.E. Gurtin "An Introduction to Continuum Mechanics" is a good book to start with. Another book is "The Mechanics and Thermodynamic of Continuous Media" by M. Silhavy. If you're interested in the geometric point of view, have a look at "Mathematical Foundations of Elasticity" by J.E. Marsden and T.J.R. Hughes. If I were you I would look at several other books and see which one I like better to start with.

When there is a map between two manifolds (if you like you can think of a manifold as a subset of the Euclidean space to make things simple) you can pull back or push forward (two-point) tensor fields. In classical continuum mechanics this map is the deformation map. Having a metric in the ambient space (or the so-called deformed configuration) you can pull it back to the reference configuration and that would be C. If you ike you can push forward the metric of the reference manifold to the ambient space (Finger tensor).

"Stress" is naturally defined in the deformed configuration. As a consequence of balance of linear momentum you can prove the existence of Cauchy stress. If you pull back the first leg of Cauchy stress you obtain the first Piola-Kirchhoff stress P. Pulling back both legs of Cauchy stress gives you the second Piola-Kirchhoff stress tensor S. Note that P and S are (energetically) conjugate to F and C, respectively (Cauchy stress is conjugate to the Lie derivative of the ambient space metric with respect to the spatial velocity field).

If you define some "stress" in the reference configuration like what some people call "configurational" or "material" stresses, you can push forward this "stress" to the ambient space, etc.

I hope you have seen my posts in this interesting thread initiated by you and the separate blog post containing some notes and a paper, on small and finite strain compatibility.

On the question of stress based formulations in nonlinear elastostatics, Janet Blume writes in her paper that a partial motivation for her work on deriving compatibility questions for the left cauchy green tensor was to see if a stress based formulation based on finite strain compatibility conditions could be achieved. The relevance of the left cauchy green tensor here is that the natural strain measure that arises in finite, *isotropic*, nonlinear elastostatics is the B tensor.

For plane deformations, Blume gives a complete characterization of the compatibility condition, which is sharply different from the corresponding condition for the right Cauchy Green field. In three dimensions, there is no necessary and sufficient condition as of date; a general sufficient condition is available in my work, but it is not explicit at all.

I will add Blume's 1989 paper to my blog post, just in case you are interested.

Thank you very much for your explanation. I am a beginner in Nonliner solid mechanics (Non-linear elasticity). I went through Cauchy's strain tensor and Almansi's strain tensor. I am trying to understand the posts. I request you to add Blume's paper.

This insightful and deep discussion stemming from a seemingly trivial question reminds me of a quote attributed to R Sachs by TJR Hughes in a set of lecture notes. The quote goes "All linear problems are trivial, all non-linear problems are impossible".

To the quote you quote, my answer would be 'more or less,' otherwise life would be too disappointing.

The fantastic utility and beauty of linear problems (and it takes a fair bit of learning to solve these), coupled with the tantalizing possibility, and occasional success, of solving some nonlinear ones is what keeps one going, I guess.

To continue the interesting discussion on compatibility of strain and the definition of stresses, Could some one explain me about the Objectivity concept in continuum mechanics ?, I read the book on the topic by Holzapfel but still vague !!